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Math Help - Evaluate the integral of 2cos(lnt)dt

  1. #1
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    Evaluate the integral of 2cos(lnt)dt

    by using substitution and then separation of variables.

    i'm lost.
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by -DQ- View Post
    by using substitution and then separation of variables.

    i'm lost.
    Substitute z := \ln t, and, therefore dt = t\cdot dz=e^z\, dz, to get

    \int 2\cos(\ln t)\, dt=\int 2\cos(z)\cdot e^z\,dz=e^z\cdot \left(\cos(z)+\sin(z)\right)+C
    =t\cdot\left(\cos(\ln t)+\sin(\ln t)\right)+C
    To determine the value of \int 2\cos(z)\cdot e^z\,dz, you do partial integration twice, and then solve for \int 2\cos(z)\cdot e^z\,dz.
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